Flexibility of planar graphs of girth at least six

Dvořák, Zdeněk; Masařík, Tomáš; Musílek, Jan; Pangrác, Ondřej

doi:10.1002/jgt.22567

Cited by 11 publications

(12 citation statements)

References 8 publications

(16 reference statements)

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“…Surprisingly, the discharging proof in [4] is quite involved compared to the easy observation that C 3 -free planar graphs are 3-degenerate, which implies 4-choosability. An analogous result holds for {C 3 , C 4 , C 5 }-free graphs, where list of size 3 are sufficient for weighted ε-flexibility and the result is tight [5].…”

Section: Introductionmentioning

confidence: 58%

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On Weak Flexibility in Planar Graphs

Lidický¹,

Masařík²,

Murphy³

et al. 2020

Preprint

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Recently, Dvořák, Norin, and Postle introduced flexibility as an extension of list coloring on graphs [JGT 19']. In this new setting, each vertex v in some subset of V (G) has a request for a certain color r(v) in its list of colors L(v). The goal is to find an L coloring satisfying many, but not necessarily all, of the requests.The main studied question is whether there exists a universal constant ε > 0 such that any graph G in some graph class C satisfies at least ε proportion of the requests. More formally, for k > 0 the goal is to prove that for any graph G ∈ C on vertex set V , with any list assignment L of size k for each vertex, and for every R ⊆ V and a request vector (r(v) : v ∈ R, r(v) ∈ L(v)), there exists an L-coloring of G satisfying at least ε|R| requests. If this is true, then C is called ε-flexible for lists of size k.Choi et al. [arXiv 20'] introduced the notion of weak flexibility, where R = V . We further develop this direction by introducing a tool to handle weak flexibility. We demonstrate this new tool by showing that for every positive integer b there exists ε(b) > 0 so that the class of planar graphs without K 4 , C 5 , C 6 , C 7 , B b is weakly ε(b)flexible for lists of size 4 (here K n , C n and B n are the complete graph, a cycle, and a book on n vertices, respectively). We also show that the class of planar graphs without K 4 , C 5 , C 6 , C 7 , B 5 is ε-flexible for lists of size 4. The results are tight as these graph classes are not even 3-colorable.

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Section: Introductionmentioning

confidence: 58%

“…The concept of ε-flexibility was introduced by Dvořák, Norin, and Postle [6]. Subsequently, it was studied for various sub-classes of planar graphs, e.g., triangle-free [4], girth six [5], or C 4 -free [9]. Graphs of bounded maximum degree were subsequently characterized in terms of flexibility [1].…”

Section: Introductionmentioning

confidence: 99%

On Weak Flexibility in Planar Graphs

Lidický¹,

Masařík²,

Murphy³

et al. 2020

Preprint

Self Cite

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show abstract

“…Apart from the original paper introducing flexibility [10], where some basic results in terms of maximum average degree were established, the main focus in flexibility research has been on planar graphs. In particular, for many subclasses G of planar graphs, there has been a vast effort to reduce the gap between the choosability of G and the list size needed for flexibility in G. As of now, some tight bounds on list sizes are known: namely, triangle-free planar graphs [8], {C 4 , C 5 }-free planar graphs [20], and {K 4 , C 5 , C 6 , C 7 , B 5 }-free 1 planar graphs [16] are flexibly 4-choosable, and planar graphs of girth 6 [9] are flexibly 3-choosable. For other subclasses G of planar graphs, only an upper-bound is known for the list size k required for G to be flexibly k-choosable [6,18]; see [6] for a comprehensive overview and a discussion of the related results.…”

Section: W(v C)mentioning

confidence: 99%

“…However, answering this question seems to be out of reach with current knowledge. Even for 2-degenerate graphs, the question seems rather tough, as it would imply the non-trivial result that planar graphs of girth 6 are flexibly 3-choosable, proven in [9], as this class of graphs is 2-degenerate.…”

Section: W(v C)mentioning

confidence: 99%

Flexible List Colorings in Graphs with Special Degeneracy Conditions

Bradshaw,

Masařík,

Stacho

2020

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For a given ε > 0, we say that a graph G is ε-flexibly k-choosable if the following holds: for any assignment L of lists of size k on V (G), if a preferred color is requested at any set R of vertices, then at least ε|R| of these requests may be satisfied by some L-coloring. We consider flexible list colorings in several graph classes with certain special degeneracy conditions. We characterize the graphs of maximum degree ∆ that are ε-flexibly ∆-choosable for some ε = ε(∆) > 0, which answers a question of Dvořák, Norin, and Postle [List coloring with requests, JGT 2019]. We also show that graphs of treewidth 2 are 1 3 -flexibly 3-choosable, answering a question of I. Choi et al. [arXiv 2020], and we give conditions for list assignments by which graphs of treewidth k are 1 k+1 -flexibly (k + 1)-choosable. We show furthermore that graphs of treedepth k are 1 k -flexibly k-choosable. Finally, we introduce a notion of flexible degeneracy, which strengthens flexible choosability, and we show that apart from a well-understood class of exceptions, three-connected non-regular graphs of maximum degree ∆ are flexibly (∆ − 1)-degenerate.

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“…This is optimal since there are triangle-free planar graphs that are not 3-choosable [5,11]. In [1] they show that planar graphs of girth at least six with an assignment of lists of size 3 are weighted ε-flexible. There is still a small gap left open since even planar graphs of girth at least 5 are 3-choosable [10].…”

Section: Introductionmentioning

confidence: 99%

Flexibility of planar graphs without 4-cycles

Masařík¹

2019

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Proper graph coloring assigns different colors to adjacent vertices of the graph. Usually, the number of colors is fixed or as small as possible. Consider applications (e.g. variants of scheduling) where colors represent limited resources and graph represents conflicts, i.e., two adjacent vertices cannot obtain the same resource. In such applications, it is common that some vertices have preferred resource(s). However, unfortunately, it is not usually possible to satisfy all such preferences. The notion called flexibility was recently defined in [Dvořák, Norin, Postle: List coloring with requests, Journal of Graph Theory 2019]. There instead of satisfying all the preferences the aim is to satisfy at least a constant fraction of the request.Recently, the structural properties of planar graphs in terms of flexibility were investigated. We continue this line of research. Let G be a planar graph with a list assignment L. Suppose a preferred color is given for some of the vertices. We prove that if G is a planar graph without 4-cycles and all lists have size at least five, then there exists an L-coloring respecting at least a constant fraction of the preferences.

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Flexibility of planar graphs of girth at least six

Cited by 11 publications

References 8 publications

On Weak Flexibility in Planar Graphs

On Weak Flexibility in Planar Graphs

Flexible List Colorings in Graphs with Special Degeneracy Conditions

Flexibility of planar graphs without 4-cycles

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